Differentiating a Linear Recursive Sequence

TL;DR

This paper demonstrates that derivatives of solutions to linear recursive sequences also satisfy linear recursions, with coefficients computable directly from the original recursion, simplifying analysis of such sequences.

Contribution

It establishes that derivatives of differentiable solutions to linear recursions follow a related linear recursion with computable coefficients, extending known operations on recursive sequences.

Findings

Derivatives satisfy a linear recursion of at most double the original order.

Coefficients of the derivative recursion are directly computed from original coefficients.

Application to derivatives of orthogonal polynomials illustrates the method.

Abstract

Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives also satisfies a homogeneous linear recursion whose order is at most double the order of original recursion. Similarly to the well-known operations that determine the elementwise sum and product of two linear recursive sequences, the coefficient functions of our recursion for the derivatives are easily computable from the original coefficient functions and their derivatives by direct manipulation of the coefficients of the characteristic polynomial of the recursion, without determining the roots. A simple application, computing linear recursions for derivatives of orthogonal polynomials, is presented.